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Prime Factorization, GCF, and LCM Practice
Fifty original questions on primes, factor trees, numeric and monomial GCF/LCM, missing values, and error analysis.
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Question 1
Explanation
The number \(1\) has only one positive factor, so it is neither prime nor composite.
Question 2
Explanation
The only positive divisors of \(2\) are \(1\) and \(2\), so it is prime.
Question 3
Explanation
The integer \(39\) has a nontrivial factor pair, so it is composite.
Question 4
Explanation
The only positive divisors of \(47\) are \(1\) and \(47\), so it is prime.
Question 5
Explanation
The integer \(91\) has a nontrivial factor pair, so it is composite.
Question 6
Explanation
Every child pair multiplies to its parent, and all leaves are prime. Grouping repeated leaves gives \(84=2^{2}\cdot3\cdot7\).
Question 7
Explanation
Every child pair multiplies to its parent, and all leaves are prime. Grouping repeated leaves gives \(126=2\cdot3^{2}\cdot7\).
Question 8
Explanation
Every child pair multiplies to its parent, and all leaves are prime. Grouping repeated leaves gives \(180=2^{2}\cdot3^{2}\cdot5\).
Question 9
Explanation
Every child pair multiplies to its parent, and all leaves are prime. Grouping repeated leaves gives \(252=2^{2}\cdot3^{2}\cdot7\).
Question 10
Explanation
Every child pair multiplies to its parent, and all leaves are prime. Grouping repeated leaves gives \(330=2\cdot3\cdot5\cdot11\).
Question 11
Explanation
Successive division by primes gives \(72=2^{3}\cdot3^{2}\). Each displayed factor is prime and their product is \(72\).
Question 12
Explanation
Successive division by primes gives \(96=2^{5}\cdot3\). Each displayed factor is prime and their product is \(96\).
Question 13
Explanation
Successive division by primes gives \(150=2\cdot3\cdot5^{2}\). Each displayed factor is prime and their product is \(150\).
Question 14
Explanation
Successive division by primes gives \(196=2^{2}\cdot7^{2}\). Each displayed factor is prime and their product is \(196\).
Question 15
Explanation
Successive division by primes gives \(360=2^{3}\cdot3^{2}\cdot5\). Each displayed factor is prime and their product is \(360\).
Question 16
Explanation
Prime-factor comparison keeps shared bases at their smaller powers. The greatest common factor is \(14\), which divides both inputs.
Question 17
Explanation
Prime-factor comparison keeps shared bases at their smaller powers. The greatest common factor is \(18\), which divides both inputs.
Question 18
Explanation
Prime-factor comparison keeps shared bases at their smaller powers. The greatest common factor is \(42\), which divides both inputs.
Question 19
Explanation
Prime-factor comparison keeps shared bases at their smaller powers. The greatest common factor is \(48\), which divides both inputs.
Question 20
Explanation
Prime-factor comparison keeps shared bases at their smaller powers. The greatest common factor is \(15\), which divides both inputs.
Question 21
Explanation
Use every prime base at the greatest exponent needed by either number. The least common multiple is \(60\), and both inputs divide it.
Question 22
Explanation
Use every prime base at the greatest exponent needed by either number. The least common multiple is \(72\), and both inputs divide it.
Question 23
Explanation
Use every prime base at the greatest exponent needed by either number. The least common multiple is \(84\), and both inputs divide it.
Question 24
Explanation
Use every prime base at the greatest exponent needed by either number. The least common multiple is \(180\), and both inputs divide it.
Question 25
Explanation
Use every prime base at the greatest exponent needed by either number. The least common multiple is \(360\), and both inputs divide it.
Question 26
Explanation
The numeric GCF is \(6\). Minimum shared powers are \(x^{2}\) and \(y^{3}\), giving \(6x^2y^3\).
Question 27
Explanation
The numeric GCF is \(12\). Minimum shared powers are \(x^{4}\) and \(y^{2}\), giving \(12x^4y^2\).
Question 28
Explanation
The numeric GCF is \(7\). Minimum shared powers are \(x^{3}\) and \(y^{2}\), giving \(7x^3y^2\).
Question 29
Explanation
The numeric GCF is \(14\). Minimum shared powers are \(x^{3}\) and \(y^{4}\), giving \(14x^3y^4\).
Question 30
Explanation
The numeric GCF is \(27\). Minimum shared powers are \(x^{2}\) and \(y^{3}\), giving \(27x^2y^3\).
Question 31
Explanation
The numeric LCM is \(36\). Maximum powers are \(x^{4}\) and \(y^{5}\), so the result is \(36x^4y^5\).
Question 32
Explanation
The numeric LCM is \(60\). Maximum powers are \(x^{6}\) and \(y^{7}\), so the result is \(60x^6y^7\).
Question 33
Explanation
The numeric LCM is \(70\). Maximum powers are \(x^{5}\) and \(y^{4}\), so the result is \(70x^5y^4\).
Question 34
Explanation
The numeric LCM is \(135\). Maximum powers are \(x^{4}\) and \(y^{6}\), so the result is \(135x^4y^6\).
Question 35
Explanation
The numeric LCM is \(48\). Maximum powers are \(x^{7}\) and \(y^{5}\), so the result is \(48x^7y^5\).
Question 36
Explanation
Keep only bases shared by both expressions and use the smaller exponent for each. This gives \(2\cdot3^2x^2\).
Question 37
Explanation
Keep only bases shared by both expressions and use the smaller exponent for each. This gives \(2^2\cdot5y^3\).
Question 38
Explanation
Keep only bases shared by both expressions and use the smaller exponent for each. This gives \(3^2\cdot7a^2\).
Question 39
Explanation
Keep only bases shared by both expressions and use the smaller exponent for each. This gives \(2^2\cdot11m^3\).
Question 40
Explanation
Keep only bases shared by both expressions and use the smaller exponent for each. This gives \(5\cdot13p^4\).
Question 41
Explanation
Prime-exponent reasoning and the identity \(ab=\operatorname{GCF}(a,b)\operatorname{LCM}(a,b)\) where applicable give \(k=3\).
Question 42
Explanation
Prime-exponent reasoning and the identity \(ab=\operatorname{GCF}(a,b)\operatorname{LCM}(a,b)\) where applicable give \(6\).
Question 43
Explanation
Prime-exponent reasoning and the identity \(ab=\operatorname{GCF}(a,b)\operatorname{LCM}(a,b)\) where applicable give \(Yes, because the GCF of 24 and 36 is 12.\).
Question 44
Explanation
Prime-exponent reasoning and the identity \(ab=\operatorname{GCF}(a,b)\operatorname{LCM}(a,b)\) where applicable give \(300\).
Question 45
Explanation
Prime-exponent reasoning and the identity \(ab=\operatorname{GCF}(a,b)\operatorname{LCM}(a,b)\) where applicable give \(36\).
Question 46
Explanation
The defining divisibility or prime-factor condition gives the correction: 1 is neither prime nor composite because it has only one positive factor.
Question 47
Explanation
The defining divisibility or prime-factor condition gives the correction: Both leaves are composite and must be split into primes.
Question 48
Explanation
The defining divisibility or prime-factor condition gives the correction: An LCM-style factor inventory.
Question 49
Explanation
The defining divisibility or prime-factor condition gives the correction: It cannot be a common factor, so it cannot be the GCF.
Question 50
Explanation
The defining divisibility or prime-factor condition gives the correction: It is not a common multiple and therefore is not the LCM.
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Questions to review
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- Question 1Prime and composite classificationEasy
- Question 2Prime and composite classificationEasy
- Question 3Prime and composite classificationEasy
- Question 4Prime and composite classificationEasy
- Question 5Prime and composite classificationEasy
- Question 6Prime factorization from a factor treeEasy
- Question 7Prime factorization from a factor treeEasy
- Question 8Prime factorization from a factor treeEasy
- Question 9Prime factorization from a factor treeEasy
- Question 10Prime factorization from a factor treeEasy
- Question 11Numeric prime factorizationEasy
- Question 12Numeric prime factorizationEasy
- Question 13Numeric prime factorizationEasy
- Question 14Numeric prime factorizationEasy
- Question 15Numeric prime factorizationEasy
- Question 16Numeric GCFMedium
- Question 17Numeric GCFMedium
- Question 18Numeric GCFMedium
- Question 19Numeric GCFMedium
- Question 20Numeric GCFMedium
- Question 21Numeric LCMMedium
- Question 22Numeric LCMMedium
- Question 23Numeric LCMMedium
- Question 24Numeric LCMMedium
- Question 25Numeric LCMMedium
- Question 26Monomial GCFMedium
- Question 27Monomial GCFMedium
- Question 28Monomial GCFMedium
- Question 29Monomial GCFMedium
- Question 30Monomial GCFMedium
- Question 31Monomial LCMMedium
- Question 32Monomial LCMMedium
- Question 33Monomial LCMMedium
- Question 34Monomial LCMMedium
- Question 35Monomial LCMMedium
- Question 36GCF of factored expressionsMedium
- Question 37GCF of factored expressionsMedium
- Question 38GCF of factored expressionsMedium
- Question 39GCF of factored expressionsMedium
- Question 40GCF of factored expressionsMedium
- Question 41Unknown GCF exponentHard
- Question 42Unknown LCM exponentHard
- Question 43GCF verificationHard
- Question 44Recover value from factorsHard
- Question 45GCF-LCM relationHard
- Question 46Factorization error analysisHard
- Question 47Factorization error analysisHard
- Question 48Factorization error analysisHard
- Question 49Factorization error analysisHard
- Question 50Factorization error analysisHard